Purpose
- To analyze spring displacement and develop a
mathematical model describing the relationship
between spring force and the distance stretched.
- Calculate the force constant of the spring
- Apply the mathematical model to determine an
expression for the potential energy of the spring.
mathematical model describing the relationship
between spring force and the distance stretched.
- Calculate the force constant of the spring
- Apply the mathematical model to determine an
expression for the potential energy of the spring.
Materials
- Spring
- masses
- Triple beam balance
- Meter stick
- Ring stand and mounting clamp for spring
- masses
- Triple beam balance
- Meter stick
- Ring stand and mounting clamp for spring
Procedure
First students will choose a spring with an appropriate force coefficient to obtain (at least) 5 data points with the weights provided.
The students will hang this spring from a ring stand and use a triple beam balance in order to level it, and attach a meter stick to the balance so that displacement can be measured.
Students will then choose 5 appropriate masses to hang from the string.
The weights chosen for this lab were: 50g 100g 150g 200g and 250g
Displacement for each weight was measured to be as follows
*Force was found using F = ma = mg = m(9.8)
The students will hang this spring from a ring stand and use a triple beam balance in order to level it, and attach a meter stick to the balance so that displacement can be measured.
Students will then choose 5 appropriate masses to hang from the string.
The weights chosen for this lab were: 50g 100g 150g 200g and 250g
Displacement for each weight was measured to be as follows
*Force was found using F = ma = mg = m(9.8)
Analysis
Since F(x) = -dU/dx, U'(x) = -F(x). F(x) = -kx
We take the antiderivative of kx (after accounting for the negatives) and get the equation for potential energy:
U = (1/2)*kx² which we know to be the equation for elastic potential energy, indicating that the spring we used in the lab was an ideal spring.
Since the spring constant is the increase in force with respect to x, it is the slope of the line F(x) or dF/dx, which was measured to be 28.481.
There were a few points that we found odd, particularly the fact that there was a y intercept that was not y=0. We assumed that this was accounted for by the force applied to the spring by the weight of the spring itself.
Another point was that we had an ideal spring, which before the class results had been thought to be a rather rare occurrence. However, every group in the class yielded a force vs displacement graph that indicated a near perfect linear relationship, in which F(x) could be presumed to be -kx.
We take the antiderivative of kx (after accounting for the negatives) and get the equation for potential energy:
U = (1/2)*kx² which we know to be the equation for elastic potential energy, indicating that the spring we used in the lab was an ideal spring.
Since the spring constant is the increase in force with respect to x, it is the slope of the line F(x) or dF/dx, which was measured to be 28.481.
There were a few points that we found odd, particularly the fact that there was a y intercept that was not y=0. We assumed that this was accounted for by the force applied to the spring by the weight of the spring itself.
Another point was that we had an ideal spring, which before the class results had been thought to be a rather rare occurrence. However, every group in the class yielded a force vs displacement graph that indicated a near perfect linear relationship, in which F(x) could be presumed to be -kx.
Conclusion
We found that a mathematical model for the relationship between spring force and displacement rather easily as it was a linear relationship that yielded very nearly
F(x) = -kx which is part of Hooke's law.
We calculated the force constant of the spring to be
k = 28.481
We took the antiderivative of the negative of our model for the relationship between spring force and displacement to find the equation for potential energy
U = (1/2)*kx²
F(x) = -kx which is part of Hooke's law.
We calculated the force constant of the spring to be
k = 28.481
We took the antiderivative of the negative of our model for the relationship between spring force and displacement to find the equation for potential energy
U = (1/2)*kx²